how to get the vertice's Horizontal and vertical location in the window?
There is a function that does the conversion from 3d space to 2d view area location_3d_to_region_2d see the DOC :
in this example, i get the 2d coordinates in the area, window and the system screen

import bpy
from bpy_extras import view3d_utils

oa = bpy.context.active_object

for area in bpy.context.screen.areas:
    if area.type == 'VIEW_3D':
        viewport = area.regions[4]
        for v in bpy.context.object.data.vertices :
            #coo in 3d space
            co_3d = oa.matrix_world * v.co
            #coo in the 3d view area (2d)
            co_2d = view3d_utils.location_3d_to_region_2d(viewport, area.spaces[0].region_3d, co_3d)
            #coo in the blender window
            co_2d_w = (co_2d[0]+viewport.x, co_2d[1]+viewport.y)
            #coo in the system screen
            co_2d_s = (co_2d_w[0]+bpy.context.window.x, co_2d_w[1]+bpy.context.window.y)
            #display
            print("\n\n3d space co :",co_3d,"\n2d in viewport co :",co_2d)              
            print("2d in window co :",co_2d_w,"\n2d system co :",co_2d_s)


NOTE :
this is what window ,area and region stands for in blender :